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WSe\(_2\)

We continue with the DFT calculation of WSe\(_2\). The three band model [LSY+13] tells us that we need to include SOC so we cannot use the standard pseudopotentials of the materialscloud. [PMC+18] Instead we take a relativistic potentials from the pslibrary compiled by Andrea Dal Corsa. [DalCorso14]

Monolayer

The input files for QE are largely the same as for graphene. The main difference is in the &SYSTEM card:

scf &SYSTEM input

&SYSTEM
  assume_isolated = '2D'
  ibrav = 4
  nat = 3
  ntyp = 2
  occupations = 'fixed'
  ecutwfc = 30
  nbnd = 36
  lspinorb = true
  noncolin = true
  a = 3.325
  c = 32
/

Description

We add two new arguments to the &SYSTEM card for the scf, nscf and bands calculations: lspinorb and noncolin. The first tells QE that at least one of pseudopotentials has SOC whereas the second tells QE to include relativistic effects. We changed occupations to fixed since WSe\(_2\) is an insulator.

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Fig. 19 QE bandstructure of WSe\(_2\) with the bands from the TB Hamiltonian of the three band model included. [LSY+13] Clearly the model is fitted to the K point and valence \(\Gamma\) point.

In Fig. 19 we plot the QE bands together with the TB bands of the three band model. [LSY+13] It is clear that this model can only describe the physics near the K point. The top two bands can not be connected to any of the QE bands. The authors show that for the calculation without SOC their model does match the band structure well. We conclude that their interpretation of SOC is too simple for an accurate model beyond the K point.

Clebsch-Gordan coefficients

For spinful calculations projwfc describes the projected orbitals using the quantum numbers for total angular momentum \(j\) and \(m_j\). Instead we wish to express the composition of the bands in terms of spin (\(m_s\)) and magnetic quantum number (\(m_l\)) which we can relate to specific types of d orbitals. There is however no one to one map between the two descriptions. In general we have:

(24)\[ |j, m_j\rangle = \sum_{m_l+m_s=m_j}C^{lsj}_{m_lm_sm_j}|l, m_l\rangle |s, m_s\rangle, \]

with \(C^{lsj}_{m_lm_sm_j}\) the Clebsch-Gordan (CG) coefficients. [Gri04]

Alas we now encounter a problem with the description of the wavefunction in the projwfc file since for \(\Psi=\sum\alpha_n\psi_n\) we only have access to the absolute value of the \(\alpha_i\). In terms of the \(|j, m_j\rangle\) this is no issue since these are all orthogonal. If we plug in equation (24) we get:

\[ \Psi = \sum_n \alpha_n|i, l, j, m_j\rangle_n = \sum_n\sum_{m_l+m_s=m_j}\alpha_nC^{lsj}_{m_lm_sm_j}|i, l, m_l, s, m_s\rangle_n, \]

with \(i\) representing the atom of the orbital.

The issue with this equation is that not all \(|i, l, m_l, s, m_s\rangle_n\) are unique and the phase of \(\alpha_n\) becomes relevant. Say we have two identical states with \(\alpha_i = a_i\text e^{i\theta_i}\) and two corresponding CG coefficients \(b_i\) (\(a_i\) and \(b_i\) are real). Now the weight of this state is:

\[ \left|a_1\text e^{i\theta_1}b_1+a_2\text e^{i\theta_2}b_2\right|^2=a_1^2b_1^2+a_2^2b_2^2+2a_1a_2b_1b_2\cos(\theta_1-\theta_2). \]

Since we have no way of knowing the difference between the two phases we can never exactly know the contribution of the state. The best we can do is to ignore the third term and take the results with a grain of salt. In the end the projections are just an indication which orbitals to use for the wannierization process.

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Fig. 20 QE bandstructure of WSe\(_2\) with projections of atomic orbitals on the wavefunctions.

One observation we can make from the projwfc results is that we have no spin splitting of the conduction band. The K point counts two valence bands:

  • The top band consists primarily of two orbitals: \(|j=2.5, l=2, m_j=2.5\rangle\) and \(|j=2.5, l=2, m_j=-2.5\rangle\) which necessarily corresponds to \(|l=2, m_l=2, m_s=0.5\rangle\) and \(|l=2, m_l=-2, m_s=-0.5\rangle\).

  • The lower band consists of two other orbitals: \(|j=1.5, l=2, m_j=1.5\rangle\) and \(|j=1.5, l=2, m_j=-1.5\rangle\) which correspond approximately to \(|l=2, m_l=2, m_s=-0.5\rangle\) and \(|l=2, m_l=-2, m_s=0.5\rangle\).

Expressing the SOC in terms of the total angular momentum this splitting makes sense:

(25)\[ \mathbf L\cdot\mathbf S = \frac12(\mathbf J^2 - \mathbf L^2 - \mathbf S^2), \]

which tells us the splitting comes from \(j\) and not \(m_s\).

Wannierization

We attempt to wannierize the bands with the same basis as the three band model. The process is not as straightforward as for graphene. Near the M point in the conduction band an unfortunate crossing creates a discontinuity in the bandstructure. On top of this the valence bands come together at the \(\Gamma\) point making it difficult to set a flat window in which the desired Wannier bands fall.

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Fig. 21 QE and W90 bands with the NNN TB Hamiltonian.

The NNN TB Hamiltonian in Fig. 21 matches the valence band well but it struggles to capture the nature of the conduction bands. We must conclude that the three chosen d orbital projections only suffice to explain the K point and the top two valence bands.

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Fig. 22 QE and W90 bands with the NNN TB Hamiltonian.

The Wannier orbitals in Fig. 22 closely resemble three d\(_{z^2}\) orbitals in the \(xy\) plane at rotations of \(2\pi/3\). The orbitals shy away from the selenide atoms and are slightly lobsided.

Graphene Bibliography